Geometry of Complex Numbers

The complex numbers  $z_1,z_2$  and  $z_3$  satisfying  $\frac{z_1-z_3}{z_2-z_3}=\frac{\mathrm{}1-i\sqrt{3}}{2},$  are the vertices of a triangle which is

If the imaginary part of $\frac{\omega }{{\omega }^{\text{'}}}$ where $\omega$ and ${\omega }^{\text{'}}$ are complex numbers be zero, show that $\omega$ and ${\omega }^{\text{'}}$ lie on a straight line passing through the origin.

For $n\in N$ let $S_n=\left\{z\in C:\left|z-3+2i\right|=\frac{n}{4}\right\}$ and $T_n=\left\{z\in C:\left|z-2+3i\right|=\frac{1}{n}\right\}$. Then the number of elements in the set $\left\{n\in N:S_n\cap T_n=\varphi \right\}$ is

On circle $\left|z-2i\right|=2$; through origin $O$ a chord $OP$ is drawn. On the tangent at $O$, a point $T$ is taken so that $OT=OP$. If $TP$ produced meet normal at $O$ to the circle at $Q$, then limiting value of $OQ$ as $P$ moves towards origin is (Note: $P$ and $T$ lie on the same side of imaginary axis)

Consider a complex number $z$ on the argand plane satisfying $\arg \left(z^2-\omega \right)=\frac{\pi }{2}+\arg \left(z^2-{\omega }^2\right)$ (where $\omega =e^{i\frac{2\pi }{3}}$ ). Then identify the correct statement(s).

Let $A\left(a\right)$ and $B\left(b\right)$ be two distinct points on the unit circle. If $Q\left(w\right)$ is the foot of perpendicular from point $P\left(z\right)$ to the line joining $A$ and $B$, then $\frac{4w}{a+b+z-ab\overline{z}}$ is

Let $A\left(z_1\right)$ be the point of intersection of curves $\arg \left(z-2+i\right)=\frac{3\pi }{4}$ and $\arg \left(z+\sqrt{3}i\right)=\frac{\pi }{3}. B\left(z_2\right)$ be the point on the curve $\arg \left(z+\sqrt{3}i\right)=\frac{\pi }{3}$ such that $\left|z_2-5\right|$ is minimum and $C\left(z_3\right)$ be the centre of the circle $\left|z-5\right|=3$. Then

If a complex number $\mathrm{z}$ lie on a circle of radius $\frac{1}{2}$ units, then the complex number $\omega =-1+4z$ will always lie on a circle of radius $k$ units, where $k$ is equal to  

If $\left|z\right|=\sqrt[4]{3} \& \omega =\frac{-1}{2}+i\frac{\sqrt{3}}{2}$, then the area of parallelogram formed by $z, \omega z, z+\omega z$ and origin is

Let $z_1, z_2, z_3$ be three complex numbers such that $arg\left(\frac{z_3-z_2}{z_1-z_2}\right)=\frac{\mathrm{\pi }}{6}$ and $\left|z_1-4\right|=\left|z_2-4\right|=\left|z_3-4\right|$, then the value of ${z_1}^2+{z_3}^2-4z_1-4z_3-z_1{z}_3+18$ is -

If ${\log }_{\frac{1}{2}}\left(\frac{\left|z-1\right|+4}{3\left|z-1\right|-2}\right)>1$, then the locus of $z$ is exterior to circle whose

For a complex number $z$, if $\lambda$ and $\mu$ are the greatest and least distance between the curves $\left|z\right|=2$ and $\left|z-5-12i\right|=2$ respectively, then the value of ${\lambda }^2+{\mu }^2$ is

The figure in the complex plane given by $10z\overline{z}-3\left(z^2+{\overline{z}}^2\right)+4i\left(z^2-{\overline{z}}^2\right)=0$, is

Let $z=x+iy$ and $w=u+iv$ be complex numbers on the unit circle such that $z^2+w^2=1$. Then the number of ordered pairs $\left(z,w\right)$ is

If $z_1=2-3\mathrm{i}$ and $z_2=-1+i,$ then the locus of a point $P$ represented by $z=x+iy$ in the Argand plane satisfying the equation $\mathit{Arg}\left(\frac{z\mathit{-}{z}_{\mathit{1}}}{z\mathit{-}{z}_{\mathit{2}}}\right)\mathit{=}\frac{\pi }{\mathit{2}}$ is

If $z, \omega z$ and $\overline{\omega }z$ are the vertices of a triangle, then the area of the triangle will be (where $\omega$ is cube root of unity) :

The locus represented by $\left|\mathrm{z}-1\right|=\left|\mathrm{z}+i\right|$ is

Locus of a point $z$ in argand plane satisfying $\left|z^2-(\overline{\mathit{z}}{)}^2\right|=|z{|}^2, \mathit{Re}(z)\ge 0, Im(z)\ge 0$ is

If a complex number $z$ satisfies $|2z+10+10i|\le 5\sqrt{3}-5$, then the maximum principal argument is